At a glance. One lesson of about 60 minutes. Students build and read stem-and-leaf plots with a key, find the mode and the median from a list and from a plot, and meet back-to-back plots.
Calculator: no calculator. Nothing here needs one. Ordering and counting to the middle is the skill being built.
What students will learn
Know
- A stem-and-leaf plot splits each value into a stem and a leaf and needs a key.
- The mode is the most frequent value; there may be none or several.
- The median is the middle value of the ordered data, or the mean of the two middle values.
Understand
- A plot with ordered leaves keeps every value and shows the shape of the data.
- The middle is only meaningful once the data is in order.
- Two data sets can be compared by their medians, side by side.
Do
- Build an ordered stem-and-leaf plot with a key, including back-to-back.
- Find the mode and the median from a list and from a plot.
- Choose the mode when the question is about the most popular value.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: choosing the right measure and applying it to lists and plots (questions 1 to 6).
- Criterion B. Investigating patterns: the waiting-time figure. Ask what happens to the median as more waits are placed and when the leaves are put in order.
- Criterion C. Communicating: reading a plot with its key correctly, and writing the median with a unit (questions 4 to 8).
- Criterion D. Applying mathematics in real-life contexts: deciding whether a mode or a median answers a shop's question (question 3) and what a bus-stop sign could honestly say.
The one thing to do with the figure
Ask for a prediction before anyone presses anything: will the most common wait be shorter or longer than the 25 minutes on the sign? Take answers from the room, then let the figure place the 21 waits.
Before the figure is on the board, read out the first five waits, 23, 31, 18, 25, 42, and ask for the middle one. Some will say 18, the third value written. Press the button and let the ordered leaves correct it. Name the rule: order first, then count to the middle.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the waiting-time figure on the board; predictions first, then place the waits. |
| 10 min | Key card, then the seedling worked example; students build the plot themselves before the steps are shown. |
| 10 min | Back-to-back plot and the even number of values; students find both medians. |
| 5 min | Question 1, the check. If many choose 9, return to the five goals and order them on the board. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Hook | Shorter: the mode is 23 minutes (4 of the 21 waits). The median is 27 minutes, longer than 25, so whether 25 is “typical” depends on which average the sign means. |
|---|---|
| Warm-up | 3; 0.09, 0.45, 0.5, 0.6; 22. |
| Worked example | Plot 2 | 8 8 8, 3 | 1 5 5 9, 4 | 1 2 4 7; mode 28 cm; median 35 cm. |
| Back-to-back example | Club X median 31, club Y median 30. |
| Faded example | 1 | 9 9 9, 2 | 2 4 7 7, 3 | 2 5, 4 | 1; mode 19; median 25.5. |
| 1. Median of an unordered list | B, 6. A means the list was not ordered; C and D read an end of the list. |
| 2. Mode from a stem-and-leaf plot | 34 cm (three times). |
| 3. Mode for a decision | 5. Sold three times. |
| 4. Median from a plot | 29 cm, the 7th value. |
| 5. Median of an even number of values | 15. Ordered: 9, 11, 14, 16, 20, 20; (14 + 16) ÷ 2. |
| 6. Reading a key with decimals | 1.8 m, the 4th of 7 values. |
| 7. Finding a missing middle value | 17. 2 × 14 − 11 = 17. |
| 8. Comparing medians in a back-to-back plot | 5 marks (69 − 64). |
| 9. Median of an even number of values from a plot | D, 41.5. A gives the position, B the mode, C only one of the middle values. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| 9 in question 1 | Took the middle of the list as written. | Ask them to write the numbers in order, then count in from both ends. |
| A, B or C in question 9 | A: the position, not the value. B: the mode. C: only the 4th value. | Ask: what are the 4th and 5th values? What number is halfway between them? |
| 4 or 3 in question 2 | Gave the leaf, or the frequency, instead of the value. | Ask: what does the key say 3 | 4 means? How many times does it appear? |
| 6 in question 3 | Gave the median when asked for the most popular size. | Ask what the shop wants to know, the middle size or the size customers choose most. |
| 34 or 7 in question 4 | 34: the mode. 7: the position of the middle, not its value. | Make them count to the 7th leaf and then read the value. |
| 14 or 16 in question 5 | Took one of the two middle values. | Ask for the number halfway between them on a number line. |
| 18 or 8 in question 6 | Ignored the stem or the decimal key. | Read the key aloud together, then read each value as a length in metres. |
| 3, 25 or 14 in question 7 | Gave the gap, the sum, or the median itself. | Draw the median halfway between the two middle values on a number line. |
Questions worth asking
- Why must the leaves be in order? The median is a position, the middle, so it depends on the order of size. Listen for “half are below, half are above”.
- Can the mode be a number that appears only twice? Yes, if no value appears three times. Ask for an example.
- Can there be two modes? None? Yes to both: two values tied for the most, or all values different.
- When is the mode a sensible answer, and when is it not? When the question is about the most popular choice (shoe size, colour). It is weak when most values appear once.
Differentiation
A way in for anyone stuck
- Give strips of paper with each value on one, so students can sort physically before they write the plot.
- Print the stems and a blank key, so only the leaves need to be written.
- Pair the first five waits with the goals check before any longer list.
Stretch
- Question 8, then: how would the medians change if one more student joined each class with the highest mark? (A new 10th value moves each median to the mean of the 5th and 6th.)
- Ask students to design a set of 7 values where the mode, the median and the largest value are all different, then trade and check.
Students learning in English as an additional language
Several of these words are everyday words with a narrower meaning here. Point to the plot as you say each one.
| Word | Plain meaning | Example |
|---|---|---|
| stem | the first digits of a value, the tens in this lesson | in 47 the stem is 4 |
| leaf | the last digit of a value, the units in this lesson | in 47 the leaf is 7 |
| key | a note that says what one entry in the plot means | 4 | 7 means 47 kg |
| mode | the value that appears most often | in 2, 5, 5, 9 the mode is 5 |
| median | the middle value when the data is in order | in 2, 5, 9 the median is 5 |
| data set | all the values collected for one question | 21 waiting times |
Sentence starters for explaining aloud or in writing:
- The mode is … because it appears … times.
- In order, the values are …, so the middle one is …
- The median is halfway between … and …, so it is …
Two things not to say
“The median is the middle number.” Say “the middle number once they are in order”, every time.
“The average is the median.” Average is a family of three measures; this unit names each one.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- ModeFind the mode of 7, 3, 7, 9, 3, 7, 5.
7, which appears three times. - MedianFind the median of 12, 5, 9, 14, 8.
9. In order: 5, 8, 9, 12, 14.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- Even numberFind the median of 30, 12, 18, 25, 40, 21.
23. In order: 12, 18, 21, 25, 30, 40, and (21 + 25) ÷ 2 = 23. - Reading a plotKey: 4 | 7 means 4.7 m. Stems 3 | 9, 4 | 1 3 3 7, 5 | 2. What is the median?
4.3 m. Six values, and the 3rd and 4th are both 4.3.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA list of 7 values has median 12 and mode 9. Write one such list.
For example 9, 9, 10, 12, 14, 15, 17. - From an earlier lessonA pie chart shows 80 students. The sector for walking has an angle of 90°. How many students walk?
20, because 90 out of 360 is a quarter and a quarter of 80 is 20.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The slider and the button work with the keyboard (tab, then the arrow keys or Enter) and by touch. Under the figure, “Show me the steps” gives the same states in words.
No device? Write the 21 values on the board and let students place them on a plot on paper.